These ideas were used in [1]} to provide the first examples of manifolds where the dimension of space of harmonic spinors depends on the underlying metric. There, it is also shown that in general the dimension of \(\mathcal {H}^{\pm }\)  only depends on the conformal class of the metric, hence in the two-dimensional case only on the Riemann surface structure. On the other hand, the parity of \(h^0(L)\)  is independent of the underlying holomorphic structure (a theta characteristic is called even/odd accordingly). This was first shown by Mumford [2]}, and reinterpreted via mod 2 index theory in the language we have just discussed by Atiyah [3]}.
